For examples of definable topologies in o-minimal structures besides the canonical “euclidean" topology, by which we mean the order topology on \(M\)  and induced product topology on \(M^n\) , see the definable (manifold) spaces studied by van den Dries [1]} and the definable metric spaces of Walsberg [2]}. For a treatment in general model theory see the work of Pillay [3]}.
